Geometry Points, Lines, Planes & Angles Part 1 2014-09-05 - PDF document

Slide 1 / 206 Slide 2 / 206 Geometry Points, Lines, Planes & Angles Part 1 2014-09-05 www.njctl.org Slide 3 / 206 Table of Contents click on the topic to go Part 1 to that section Introduction to Geometry Points and Lines Planes Congruence, Distance and Length Constructions and Loci Part 2 Angles Congruent Angles Angles & Angle Addition Postulate Protractors Special Angle Pairs Proofs Special Angles Angle Bisectors Locus & Angle Constructions Angle Bisectors & Constructions

Slide 4 / 206 Introduction to Geometry Return to Table of Contents Slide 5 / 206 The Origin of Geometry About 10,000 years ago much of North Africa was fertile farmland. The area around the Nile river was too marshy for agriculture, so it was sparsely populated. Slide 6 / 206 The Origin of Geometry But over thousands of years the climate changed, and most of North African became desert. The banks of the Nile became prime farmland.

Slide 7 / 206 The Origin of Geometry The land along the Nile became crowded with people. Farming was done on the land near the river because it had: Water for irrigation · Fertile soil due to annual flooding, which deposited silt from · upriver. But, since the land flooded each year, how could they keep track of who owned which land? Slide 8 / 206 Egyptian Geometry About 4000 years ago an Egyptian pharaoh, Sesostris, is said to have invented geometry in order to keep track of the land and tax it's owners. Reestablishing land ownership after each annual flood required a practical geometry. "Geo" means Earth and "metria" means measure, so geometry meant to measure land. Slide 9 / 206 Land Boundaries Lab You know more geometry than the Egyptians knew 4000 years ago, so let's do a lab to see how you would solve this problem. You'll work in groups and each group will solve this problem before we move on to how the Greek's built on the Egyptian solution.

Slide 10 / 206 Land Boundaries Lab Pre- Flood Boundary Map Before the annual flood A of the Nile three plots of land might be as shown. Plot 1 The orange dots are to B indicate stakes that were placed above the Plot 2 flood level. C D The stakes would remain in the same location from year to year. Plot 3 E Slide 11 / 206 Land Boundaries Lab Afterwards, only the stakes above Before flooding, three plots of the flood level remained, and the land might be look like these. river had moved in its course. Pre- Flood Boundary Map Post-Flood Map of River and Markers A A Plot 1 B B Plot 2 C C D D Plot 3 E E Slide 12 / 206 Land Boundaries Lab The pharaoh had to: Reestablish new boundaries so farmers knew which land to farm. · Adjust the taxes to match the new amount of land owned. · The Egyptians only had stakes and rope, you only have tape and string.

Slide 13 / 206 Land Boundaries Lab Post-Flood Map of River and Markers After the flood, the pharaoh would send out A geometers with ropes that had been used to measure each plot of land in prior years. B How did they do it? C (You can't use the edges D of the paper or rulers because these were open fields of great size.) E Slide 14 / 206 Egyptian Geometry Egyptian mathematics was very Teacher Notes practical. What practical applications do you think the Egyptians used mathematics for? They did not develop abstract mathematics. That was left to the Greeks, who built upon what they had learned from the Egyptians, Babylonians and others. Slide 15 / 206 Greek Geometry The Greeks developed an approach to thinking about these earth measures that allowed them to be generalized. They kept their assumptions to the minimum, and showed how all else followed from those assumptions. Those assumptions are called definitions, postulates and axioms. That analytical thinking became the logic that allows us to not only measure land, but also measure the validity of ideas.

Slide 16 / 206 Euclidean Geometry Euclid's book, The Elements , summarized the results of Greek geometry: Euclidean Geometry. Euclidean geometry is the basis of much of western mathematics, philosophy and science. It also represents a great place to learn that type of thinking. Slide 17 / 206 Euclidean Geometry Euclidean Geometry dates prior to 400 BC. That makes it about 1000 years older than algebra, and about 2000 years older than calculus. The fact that it is still taught in much the way it was more than 2000 years ago tells us what about Euclid's ideas? Slide 18 / 206 Euclidean Geometry "Let none who are ignorant of geometry enter here." This statement was posted above Plato's Academy, in ancient Athens, about 2500 years ago. This renaissance painting by Raphael depicts that academy.

Slide 19 / 206 Euclidean Geometry When the Roman empire declined, and then fell, about 1800 years ago, most of the writings of Greek civilization were lost. This included most of the plays, histories, philosophical, scientific and mathematical works of that era, including The Elements by Euclid. These works were not purposely destroyed, but deteriorated with age as there was no central government to maintain them. Slide 20 / 206 Euclidean Geometry Euclidean Geometry was lost to Europe for a 1000 years. But, it continued to be used and developed in the Islamic world. In the 1400's, these ideas were reintroduced to Europe. These, and other rediscovered works, led to the European Renaissance, which lasted several centuries, beginning in the 1400's. Slide 21 / 206 Euclidean Geometry Much of the thinking of modern science and mathematics developed from the rediscovery of Euclid's Elements. The thinking that underlies Euclidean Geometry has held up very well. Many still believe it is the best introduction to analytical thinking.

Slide 22 / 206 Euclidean Geometry About 100 years ago, Charles Dodgson, the Oxford geometer who wrote Alice in Wonderland, under the name Lewis Carroll, argued Euclid was still the best way to understand mathematical thinking. Slide 23 / 206 Euclidean Geometry Geometry is used directly in many tasks such as measuring lengths, areas and volumes; surveying land, designing optics, etc. Geometry underlies much of science, technology, engineering and mathematics (STEM). Slide 24 / 206 Euclidean Geometry This course will use the basic thinking developed by Euclid. We will attempt to make clear and distinguish between: What we have assumed to be true, and cannot prove · What follows from what we have previously assumed or proven · That is the reasoning that makes geometric thinking so valuable. Always question every idea that's presented, that's what Euclid and those who invented geometry would have wanted.

Slide 25 / 206 Euclidean Geometry This also represents a path to logical thinking, which British philosopher Bertrand Russell showed is identical to mathematical thinking. Click on the image to watch a short video of Bertrand Russell's message to the future which was filmed in 1959. Did you hear anything that sounded familiar? What was it? Slide 26 / 206 Euclidean Geometry Euclid's assumptions are axioms , postulates and definitions . You won't be expected to memorize them, but to use them to develop further understanding. Major ideas which are proven are called Theorems . Ideas that easily follow from a theorem are called Corollaries . Slide 27 / 206 Euclidean Geometry The five axioms are very general, apply to the entire course, and do not depend on the definitions or postulates, so we'll review them in this unit. The postulates and definitions are related to specific topics, so we will introduce them as required. Also, additional modern terms which you will need to know will be introduced as needed.

Slide 28 / 206 Euclid's Axioms (Common Understandings) Euclid called his axioms "Common Understandings." They seem so obvious to us now, and to him then, that the fact that he wrote them down as his assumptions reflects how carefully he wanted to make clear his thinking. He didn't want to assume even the most obvious understandings without indicating that he was doing just that. Slide 29 / 206 Euclid's Axioms (Common Understandings) This careful rigor is what led to this approach changing the world. Great breakthroughs in science, mathematics, engineering, business, etc. are made by people who question what seems obviously true...but turns out to not always be true. Without recognizing the assumptions you are making, you're not able to question them...and, sometimes, not able to move beyond them. Slide 30 / 206 Euclid's First Axiom Things which are equal to the same thing are also equal to one another. For example: if I know that Tom and Bob are the same height, and I know that Bob and Sarah are the same height...what other conclusion can I come to? Tom Bob Sarah

Slide 31 / 206 Euclid's Second Axiom If equals are added to equals, the whole are equal. For example, if you and I each have the same amount of money, let's say $20, and we each earn the same additional amount, let's say $2, then we still each have the same total amount of money as each other, in this case $22. Slide 32 / 206 Euclid's Third Axiom If equals be subtracted from equals, the remainders are equal. This is just like the second axiom. Come up with an example on your own. Look back at the second axiom if you need a hint. Slide 33 / 206 Euclid's Fourth Axiom Things which coincide with one another are equal to one another. For example, if I lay two pieces of wood side by side and both ends and all the points in between line up, I would say they have equal lengths.